Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$\frac{\cos{\left(x \right)}}{\left|{x}\right|} - \frac{\sin{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -86.3822220347287$$
$$x_{2} = -4.49340945790906$$
$$x_{3} = -42.3879135681319$$
$$x_{4} = -32.9563890398225$$
$$x_{5} = -67.5294347771441$$
$$x_{6} = -20.3713029592876$$
$$x_{7} = 23.519452498689$$
$$x_{8} = 61.2447302603744$$
$$x_{9} = 7.72525183693771$$
$$x_{10} = -92.6661922776228$$
$$x_{11} = 14.0661939128315$$
$$x_{12} = -45.5311340139913$$
$$x_{13} = 48.6741442319544$$
$$x_{14} = 70.6716857116195$$
$$x_{15} = 64.3871195905574$$
$$x_{16} = 95.8081387868617$$
$$x_{17} = 36.1006222443756$$
$$x_{18} = -29.811598790893$$
$$x_{19} = -76.9560263103312$$
$$x_{20} = -10.9041216594289$$
$$x_{21} = 98.9500628243319$$
$$x_{22} = 76.9560263103312$$
$$x_{23} = 45.5311340139913$$
$$x_{24} = 39.2444323611642$$
$$x_{25} = -98.9500628243319$$
$$x_{26} = -89.5242209304172$$
$$x_{27} = -61.2447302603744$$
$$x_{28} = 17.2207552719308$$
$$x_{29} = 92.6661922776228$$
$$x_{30} = 4.49340945790906$$
$$x_{31} = 54.9596782878889$$
$$x_{32} = -394.267341680887$$
$$x_{33} = -48.6741442319544$$
$$x_{34} = 67.5294347771441$$
$$x_{35} = -7.72525183693771$$
$$x_{36} = -17.2207552719308$$
$$x_{37} = 86.3822220347287$$
$$x_{38} = 32.9563890398225$$
$$x_{39} = -26.6660542588127$$
$$x_{40} = 26.6660542588127$$
$$x_{41} = 80.0981286289451$$
$$x_{42} = 108.375719651675$$
$$x_{43} = -95.8081387868617$$
$$x_{44} = 20.3713029592876$$
$$x_{45} = -83.2401924707234$$
$$x_{46} = 10.9041216594289$$
$$x_{47} = 83.2401924707234$$
$$x_{48} = -202.627791039417$$
$$x_{49} = 89.5242209304172$$
$$x_{50} = 29.811598790893$$
$$x_{51} = 58.1022547544956$$
$$x_{52} = -54.9596782878889$$
$$x_{53} = -64.3871195905574$$
$$x_{54} = -39.2444323611642$$
$$x_{55} = -14.0661939128315$$
$$x_{56} = -70.6716857116195$$
$$x_{57} = -73.8138806006806$$
$$x_{58} = 73.8138806006806$$
$$x_{59} = -36.1006222443756$$
$$x_{60} = -58.1022547544956$$
$$x_{61} = 42.3879135681319$$
$$x_{62} = -51.8169824872797$$
$$x_{63} = -23.519452498689$$
$$x_{64} = 51.8169824872797$$
$$x_{65} = -80.0981286289451$$
Signos de extremos en los puntos:
(-86.38222203472871, 0.0115756804584678)
(-4.493409457909064, 0.217233628211222)
(-42.38791356813192, 0.0235850682290164)
(-32.956389039822476, -0.0303291711863103)
(-67.52943477714412, 0.0148067339465492)
(-20.37130295928756, -0.0490296240140742)
(23.519452498689006, -0.0424796169776126)
(61.2447302603744, -0.0163257593209978)
(7.725251836937707, 0.128374553525899)
(-92.66619227762284, 0.0107907938495342)
(14.066193912831473, 0.0709134594504622)
(-45.53113401399128, -0.0219576982284824)
(48.674144231954386, -0.0205404540417537)
(70.6716857116195, 0.0141485220648664)
(64.38711959055742, 0.0155291838074613)
(95.8081387868617, 0.0104369581345658)
(36.10062224437561, -0.0276897323011492)
(-29.81159879089296, 0.0335251350213988)
(-76.95602631033118, -0.0129933369870427)
(-10.904121659428899, 0.0913252028230577)
(98.95006282433188, -0.010105591736504)
(76.95602631033118, 0.0129933369870427)
(45.53113401399128, 0.0219576982284824)
(39.24443236116419, 0.0254730530928808)
(-98.95006282433188, 0.010105591736504)
(-89.52422093041719, -0.0111694646341736)
(-61.2447302603744, 0.0163257593209978)
(17.22075527193077, -0.0579718023461539)
(92.66619227762284, -0.0107907938495342)
(4.493409457909064, -0.217233628211222)
(54.959678287888934, -0.0181921463218031)
(-394.26734168088706, 0.00253634191261283)
(-48.674144231954386, 0.0205404540417537)
(67.52943477714412, -0.0148067339465492)
(-7.725251836937707, -0.128374553525899)
(-17.22075527193077, 0.0579718023461539)
(86.38222203472871, -0.0115756804584678)
(32.956389039822476, 0.0303291711863103)
(-26.666054258812675, -0.0374745199939312)
(26.666054258812675, 0.0374745199939312)
(80.09812862894512, -0.012483713321779)
(108.37571965167469, 0.00922676625078197)
(-95.8081387868617, -0.0104369581345658)
(20.37130295928756, 0.0490296240140742)
(-83.2401924707234, -0.0120125604820527)
(10.904121659428899, -0.0913252028230577)
(83.2401924707234, 0.0120125604820527)
(-202.62779103941682, -0.00493509709208483)
(89.52422093041719, 0.0111694646341736)
(29.81159879089296, -0.0335251350213988)
(58.10225475449559, 0.0172084874716279)
(-54.959678287888934, 0.0181921463218031)
(-64.38711959055742, -0.0155291838074613)
(-39.24443236116419, -0.0254730530928808)
(-14.066193912831473, -0.0709134594504622)
(-70.6716857116195, -0.0141485220648664)
(-73.81388060068065, 0.01354634434514)
(73.81388060068065, -0.01354634434514)
(-36.10062224437561, 0.0276897323011492)
(-58.10225475449559, -0.0172084874716279)
(42.38791356813192, -0.0235850682290164)
(-51.81698248727967, -0.019295099487588)
(-23.519452498689006, 0.0424796169776126)
(51.81698248727967, 0.019295099487588)
(-80.09812862894512, 0.012483713321779)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = -32.9563890398225$$
$$x_{2} = -20.3713029592876$$
$$x_{3} = 23.519452498689$$
$$x_{4} = 61.2447302603744$$
$$x_{5} = -45.5311340139913$$
$$x_{6} = 48.6741442319544$$
$$x_{7} = 36.1006222443756$$
$$x_{8} = -76.9560263103312$$
$$x_{9} = 98.9500628243319$$
$$x_{10} = -89.5242209304172$$
$$x_{11} = 17.2207552719308$$
$$x_{12} = 92.6661922776228$$
$$x_{13} = 4.49340945790906$$
$$x_{14} = 54.9596782878889$$
$$x_{15} = 67.5294347771441$$
$$x_{16} = -7.72525183693771$$
$$x_{17} = 86.3822220347287$$
$$x_{18} = -26.6660542588127$$
$$x_{19} = 80.0981286289451$$
$$x_{20} = -95.8081387868617$$
$$x_{21} = -83.2401924707234$$
$$x_{22} = 10.9041216594289$$
$$x_{23} = -202.627791039417$$
$$x_{24} = 29.811598790893$$
$$x_{25} = -64.3871195905574$$
$$x_{26} = -39.2444323611642$$
$$x_{27} = -14.0661939128315$$
$$x_{28} = -70.6716857116195$$
$$x_{29} = 73.8138806006806$$
$$x_{30} = -58.1022547544956$$
$$x_{31} = 42.3879135681319$$
$$x_{32} = -51.8169824872797$$
Puntos máximos de la función:
$$x_{32} = -86.3822220347287$$
$$x_{32} = -4.49340945790906$$
$$x_{32} = -42.3879135681319$$
$$x_{32} = -67.5294347771441$$
$$x_{32} = 7.72525183693771$$
$$x_{32} = -92.6661922776228$$
$$x_{32} = 14.0661939128315$$
$$x_{32} = 70.6716857116195$$
$$x_{32} = 64.3871195905574$$
$$x_{32} = 95.8081387868617$$
$$x_{32} = -29.811598790893$$
$$x_{32} = -10.9041216594289$$
$$x_{32} = 76.9560263103312$$
$$x_{32} = 45.5311340139913$$
$$x_{32} = 39.2444323611642$$
$$x_{32} = -98.9500628243319$$
$$x_{32} = -61.2447302603744$$
$$x_{32} = -394.267341680887$$
$$x_{32} = -48.6741442319544$$
$$x_{32} = -17.2207552719308$$
$$x_{32} = 32.9563890398225$$
$$x_{32} = 26.6660542588127$$
$$x_{32} = 108.375719651675$$
$$x_{32} = 20.3713029592876$$
$$x_{32} = 83.2401924707234$$
$$x_{32} = 89.5242209304172$$
$$x_{32} = 58.1022547544956$$
$$x_{32} = -54.9596782878889$$
$$x_{32} = -73.8138806006806$$
$$x_{32} = -36.1006222443756$$
$$x_{32} = -23.519452498689$$
$$x_{32} = 51.8169824872797$$
$$x_{32} = -80.0981286289451$$
Decrece en los intervalos
$$\left[98.9500628243319, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -202.627791039417\right]$$